Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $p \neq 0$. $q = \dfrac{27p - 72}{-4} \div \dfrac{6p^2 - 16p}{8p} $
Dividing by an expression is the same as multiplying by its inverse. $q = \dfrac{27p - 72}{-4} \times \dfrac{8p}{6p^2 - 16p} $ When multiplying fractions, we multiply the numerators and the denominators. $q = \dfrac{ (27p - 72) \times 8p } { -4 \times (6p^2 - 16p) } $ $ q = \dfrac {8p \times 9(3p - 8)} {-4 \times 2p(3p - 8)} $ $ q = \dfrac{72p(3p - 8)}{-8p(3p - 8)} $ We can cancel the $3p - 8$ so long as $3p - 8 \neq 0$ Therefore $p \neq \dfrac{8}{3}$ $q = \dfrac{72p \cancel{(3p - 8})}{-8p \cancel{(3p - 8)}} = -\dfrac{72p}{8p} = -9 $